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This question comes from observation that there are no known half-lives in range;
$1\times 10^{-10}$ seconds to $1\times 10^{-21}$ seconds.
(Except Beryllium-8, which has a half-life of o $7\times 10^{-18}$ seconds.)

As the isotopes are mostly produced with a neutron Flux, which practically means, that neutrons are colliding with a certain velocity to the target particle, I became an idea what this actually means.

A neutron is very similar to a proton, and Proton diameter is said to be $0.84\times 10^{-15}$ meters. If I calculate with typical Neutron speed which is present in Nuclear-fission; $1960000$ m/s, it would take at least $4\times 10^{-22}$ second, for a Neutron to travel away from its position which is farther away than it's own size.

But as radioactive decay is happening all the time, more presenting speed would be that of Thermal Neutrons $2200$ m/s. This means that Neutron needs $4\times 10^{-19}$ second to change is position more than its size is.

Calculation with the speed of ultra-cold Neutrons; speed $<200$ m/s, gives for a time $4\times 10^{-18}$ seconds. This simple rule would mean that only Beryllium-8 would have long-enough half-life to be an independent nucleus, compared to pure neutron collision. But looking this isotope, shows, that it decays with $\alpha$-decay. Which in this case means that it would split in two equal Helium-4 nucleus.

QUESTION;
Have such a theoretical limit for an independent isotope established? ..And if yes, how is it explained that some isotopes with just $23\times 10^{-24}$ like Hydrogen-7 are considered to be something else than just colliding neutrons?

Rick
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Jokela
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2 Answers2

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First of all, the Wikipedia article you linked to is not complete and doesn't claim to be. (As of the time of writing, the header says "This list is incomplete; you can help by expanding it.") For example, magnesium-19 has a half-life of about $4 \times 10^{-12}$ s as measured by Mukha et al.

The main reason we can't observe isotopes with femtosecond half-lives is experimental. Very short-lived isotopes are normally detected by smashing nuclei into pieces using high-energy collisions, then tracking the products using magnetic separation, using an instrument like the BigRIPS separator. However, the speed of light becomes a limiting factor: in $10^{-15}$ s, a nucleus cannot travel more than 300 nm. Accurately tracking nuclei over such short distances is very difficult (even the CMS tracker at LHC is only accurate to around 10 $\mu$m). In theory, relativistic time dilation might help a little, but on the other hand, isotope separation becomes harder for very fast-moving, short-lived nuclei because magnetic fields barely affect their motion.

But to answer your question, the theoretical limit on very short half-lives is the fact that information cannot travel faster than the speed of light. Obviously, a nucleus must be bound before we can discuss its half-life. But it makes no sense to say that a nucleus is bound if it decays before all of its constituent protons and neutrons "know" that it has formed. Therefore, since nuclei must be larger than the radius of a proton, a hard lower limit on nuclear half-lives is $r_\text{proton} / c \approx 2.9 \times 10^{-24}$ s. (Not coincidentally, the strong nuclear interaction has a characteristic time scale on the same order of magnitude.) Indeed, the half-lives of hydrogen-6 and hydrogen-7 are within an order of magnitude or two of this limit.

Thorondor
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A decay time of ${10^{-24}}$ s is characteristic for a process mediated by the strong nuclear interaction.

Also an intuitive picture like colliding nucleons is not a viable approach here. In particular you should not think of the neutron as a sphere with a certain diameter.

polwel
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